Q: What is the prime factorization of the number 10,418,688?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 17 x 19
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 7, 17, 19 }
  • Written in exponential form: 29 x 32 x 71 x 171 x 191

Why is the prime factorization of 10,418,688 written as 29 x 32 x 71 x 171 x 191?

What is prime factorization?

Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

Finding the prime factors of 10,418,688

To find the prime factors, you start by dividing the number by the first prime number, which is 2. If there is not a remainder, meaning you can divide evenly, then 2 is a factor of the number. Continue dividing by 2 until you cannot divide evenly anymore. Write down how many 2's you were able to divide by evenly. Now try dividing by the next prime factor, which is 3. The goal is to get to a quotient of 1.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 10,418,688 by 2

10,418,688 ÷ 2 = 5,209,344 - No remainder! 2 is one of the factors!
5,209,344 ÷ 2 = 2,604,672 - No remainder! 2 is one of the factors!
2,604,672 ÷ 2 = 1,302,336 - No remainder! 2 is one of the factors!
1,302,336 ÷ 2 = 651,168 - No remainder! 2 is one of the factors!
651,168 ÷ 2 = 325,584 - No remainder! 2 is one of the factors!
325,584 ÷ 2 = 162,792 - No remainder! 2 is one of the factors!
162,792 ÷ 2 = 81,396 - No remainder! 2 is one of the factors!
81,396 ÷ 2 = 40,698 - No remainder! 2 is one of the factors!
40,698 ÷ 2 = 20,349 - No remainder! 2 is one of the factors!
20,349 ÷ 2 = 10,174.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
20,349 ÷ 3 = 6,783 - No remainder! 3 is one of the factors!
6,783 ÷ 3 = 2,261 - No remainder! 3 is one of the factors!
2,261 ÷ 3 = 753.6667 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
2,261 ÷ 5 = 452.2 - This has a remainder. 5 is not a factor.
2,261 ÷ 7 = 323 - No remainder! 7 is one of the factors!
323 ÷ 7 = 46.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
323 ÷ 11 = 29.3636 - This has a remainder. 11 is not a factor.
323 ÷ 13 = 24.8462 - This has a remainder. 13 is not a factor.
323 ÷ 17 = 19 - No remainder! 17 is one of the factors!
19 ÷ 17 = 1.1176 - There is a remainder. We can't divide by 17 evenly anymore. Let's try the next prime number
19 ÷ 19 = 1 - No remainder! 19 is one of the factors!

The orange divisor(s) above are the prime factors of the number 10,418,688. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7 x 17 x 19 = 10,418,688. It can also be written in exponential form as 29 x 32 x 71 x 171 x 191.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 10,418,688.

10,418,688
Factor Arrows
25,209,344
Factor Arrows
22,604,672
Factor Arrows
21,302,336
Factor Arrows
2651,168
Factor Arrows
2325,584
Factor Arrows
2162,792
Factor Arrows
281,396
Factor Arrows
240,698
Factor Arrows
220,349
Factor Arrows
36,783
Factor Arrows
32,261
Factor Arrows
7323
Factor Arrows
1719

More Prime Factorization Examples

10,418,68610,418,68710,418,68910,418,690
21 x 4571 x 11,3991531 x 196,579110,418,689121 x 51 x 1,041,8691

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